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Interpreting a Finitary Pi-calculus in Differential Interaction Nets
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Thomas Ehrhard, Olivier Laurent

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Lecture Notes in Computer Science
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We propose and study a translation of a pi-calculus without sums nor recursion into an untyped version of differential interaction nets. We define a transition system of labeled processes and a transition system of labeled differential interaction nets. We prove that our translation from processes to nets is a bisimulation between these two transition systems. This shows that differential interaction nets are sufficiently expressive for representing concurrency and mobility, as formalized by
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... pi-calculus. Our study will concern essentially a replication-free fragment of the pi-calculus, but we shall also give indications on how to deal with a restricted form of replication. (T. Ehrhard). 1 The only non symmetric rule of DiLL is promotion. Finding a symmetric version thereof seems to be a rather challenging task! 0890-5401/$ -see front matter π-calculus previously introduced by Berger et al. [6] . This translation has two features which can be considered as slight defects: it accepts only replicable receivers and it is not really modular (the parallel composition of two processes cannot be described as a combination of the corresponding nets). One should mention here that translations of the π-calculus into nets of various kinds, subject to local reduction relations, have been provided by several authors (cf. the work of Laneve et al. on solo diagrams [22], of Beffara and Maurel [7], of Jensen and Milner on bigraphs [20], of Mazza [24] on multiport interaction nets, etc.). One should also mention the early work of Honda and Yoshida [19] which introduces a system of combinators for interpreting a process algebra. These combinators have connections with Lafont's interaction nets; just like multiport interaction nets and solo diagrams, this system seems however to lack the main feature of interaction nets, namely (strong) confluence. Moreover, as far as we know, these approaches have no clear logical grounds nor simple denotational semantics. Indeed, the fact that DINs have a denotational semantics, together with the translation we propose, suggests to interpret the π-calculus in DINs' denotational models and to study the induced equivalence of processes. This approach will be developed in further work. It should be observed moreover that the denotational models of DINs' are also models of the lambda-calculus, suggesting natural combinations between concurrent programming (as modeled in DINs) and functional programming. Principle of the translation The purpose of the present paper is to continue this line of ideas, using more systematically the new structures introduced by DINs. The first key decision we made, guided by the structure of the typical cocontraction/contraction cut intended to interpret parallel composition, was of associating with each free name of a process not one, but two free ports in the corresponding differential interaction net. One of these ports will have a !-type (positive type) and will have to be considered as the input port of the corresponding name for this process, and the other one will have a ?-type (negative type) and will be considered as an output port. We discovered structures which allow one to combine these pairs of wires for interpreting parallel composition and called them communication areas: they can be seen as complete graphs between vertices made of pairs of contraction cells (marked by a "?" symbol) and cocontraction cells (marked by a "!" symbol), connected by edges which are pairs of wires. An example of such a structure, with three vertices, is given in Fig. 1 . Output and input prefixes will be interpreted using dereliction and codereliction, as well as the multiplicative connectives. Content We first introduce differential linear logic, presented as a sequent calculus, and then differential interaction nets. These nets are typed with the recursive typing system introduced by Danos and Regnier [28] (which corresponds to the untyped lambda-calculus) for avoiding the appearance of non reducible configurations. To simplify the presentation, these nets use only a restricted form of the promotion rule of linear logic, which is sufficient for interpreting a replication-free version of the π-calculus, as well as a restricted form of replication. In this setting, we define a "toolbox", a collection of nets that we shall combine for interpreting processes, and a few associated reductions, derived from the basic reduction rules of differential interaction nets. We organize reduction rules of nets as a labeled transition system, whose vertices are nets, and where the transitions correspond to dereliction/codereliction reductions. Then we define a process algebra which is a polyadic π-calculus, without replication and without sums. We specify the operational semantics of this calculus by means of an abstract machine inspired

doi:10.1007/978-3-540-74407-8_23
fatcat:fpb4eh57tbekndoxxqgdr6k2si